A Super-High-Efficiency Algorithm for the Calculation of the Correlation Integral

Author(s)
Zhuping Gong

Affiliation(s)

School of Business and Administration, South China University of Technology, Guangzhou, China.

School of Business and Administration, South China University of Technology, Guangzhou, China.

ABSTRACT

When the chaotic characteristics of manufacturing quality level are studied, it is not practical to use chaotic methods because of the low speed of calculating the correlation integral. The original algorithm used to calculate the correlation integral is studied after a computer hardware upgrade. The result is that calculation of the correlation integral can be sped up only by improving the algorithm. This is accomplished by changing the original algorithm in which a single distance threshold-related correlation integral is obtained from one traversal of all distances between different vectors to a high-efficiency algorithm in which all of the distance threshold-related correlation integrals are obtained from one traversal of all of the distances between different vectors. For a time series with 3000 data points, this high-efficiency algorithm offers a 3.7-fold increase in speed over the original algorithm. Further study of the high-efficiency algorithm leads to the development of a super-high-efficiency algorithm, which is accomplished by changing the original and high-efficiency algorithms, in which the add-one operation of the Heaviside function is executed n times, such that the execution of the add-one operation occurs only once. The super-high-efficiency algorithm results in increases in the calculation speed by up to 109 times compared with the high-efficiency algorithm and by approximately 404 times compared with the original algorithm. The calculation speed of the super-high-efficiency algorithm is suitable for practical use with the chaotic method.

When the chaotic characteristics of manufacturing quality level are studied, it is not practical to use chaotic methods because of the low speed of calculating the correlation integral. The original algorithm used to calculate the correlation integral is studied after a computer hardware upgrade. The result is that calculation of the correlation integral can be sped up only by improving the algorithm. This is accomplished by changing the original algorithm in which a single distance threshold-related correlation integral is obtained from one traversal of all distances between different vectors to a high-efficiency algorithm in which all of the distance threshold-related correlation integrals are obtained from one traversal of all of the distances between different vectors. For a time series with 3000 data points, this high-efficiency algorithm offers a 3.7-fold increase in speed over the original algorithm. Further study of the high-efficiency algorithm leads to the development of a super-high-efficiency algorithm, which is accomplished by changing the original and high-efficiency algorithms, in which the add-one operation of the Heaviside function is executed n times, such that the execution of the add-one operation occurs only once. The super-high-efficiency algorithm results in increases in the calculation speed by up to 109 times compared with the high-efficiency algorithm and by approximately 404 times compared with the original algorithm. The calculation speed of the super-high-efficiency algorithm is suitable for practical use with the chaotic method.

Cite this paper

Gong, Z. (2015) A Super-High-Efficiency Algorithm for the Calculation of the Correlation Integral.*Journal of Data Analysis and Information Processing*, **3**, 128-135. doi: 10.4236/jdaip.2015.34013.

Gong, Z. (2015) A Super-High-Efficiency Algorithm for the Calculation of the Correlation Integral.

References

[1] Gong, Z.P. (2010) The Calculating Method of the Average Period of Chaotic Time Series. Systems Engineering, 28, 111-113. (In Chinese)

[2] Gong, Z.P. (2011) Comparison of the Calculating Method of Delay Time in the Reconstructed Phase Space of Manufacturing Quality Information System. Systems Engineering, 29, 81-85. (In Chinese)

[3] Gong, Z.P. (2012) The Chaotic Characteristic of the Variation of Manufacturing Quality Level Based on Time Series Analysis. Systems Engineering, 30, 38-42. (In Chinese)

[4] Grassberger, P. and Procaccia, I. (1983) Measuring the Strangeness of Strange Attractors. Physica D: Nonlinear Phenomena, 9, 189-208. http://dx.doi.org/10.1016/0167-2789(83)90298-1

[5] Kantz, H. and Schreiber, T. (1997) Nonlinear Time Series Analysis. Cambridge University Press, Cambridge.

[6] Kim, H.S., Eykholt, R. and Salas, J.D. (1999) Nonlinear Dynamics, Delay Times, and Embedding Windows. Physica D: Nonlinear Phenomena, 127, 48-60. http://dx.doi.org/10.1016/S0167-2789(98)00240-1

[1] Gong, Z.P. (2010) The Calculating Method of the Average Period of Chaotic Time Series. Systems Engineering, 28, 111-113. (In Chinese)

[2] Gong, Z.P. (2011) Comparison of the Calculating Method of Delay Time in the Reconstructed Phase Space of Manufacturing Quality Information System. Systems Engineering, 29, 81-85. (In Chinese)

[3] Gong, Z.P. (2012) The Chaotic Characteristic of the Variation of Manufacturing Quality Level Based on Time Series Analysis. Systems Engineering, 30, 38-42. (In Chinese)

[4] Grassberger, P. and Procaccia, I. (1983) Measuring the Strangeness of Strange Attractors. Physica D: Nonlinear Phenomena, 9, 189-208. http://dx.doi.org/10.1016/0167-2789(83)90298-1

[5] Kantz, H. and Schreiber, T. (1997) Nonlinear Time Series Analysis. Cambridge University Press, Cambridge.

[6] Kim, H.S., Eykholt, R. and Salas, J.D. (1999) Nonlinear Dynamics, Delay Times, and Embedding Windows. Physica D: Nonlinear Phenomena, 127, 48-60. http://dx.doi.org/10.1016/S0167-2789(98)00240-1